Period-Resolvent Recurrent Layer
Implementation & benchmark of arXiv:2607.24857 — Arithmetic Landscape Functions of a Discrete Cat Map
Source paper: Arithmetic Landscape Functions of a Discrete Cat Map arXiv:2607.24857 ⓘ · analyzed Aug 31, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Construct a recurrent or graph-neural layer on a finite state space with a known bijection T, such as a modular cat map, and use the diagonal resolvent gain (1 − α^kx)^−1 as a state-dependent self-return or memory coefficient. States on short periodic orbits receive larger amplification, while long-period states receive weaker amplification, producing deterministic localization without learned disorder.
Formulas
Mathematical statement
Let X be a finite state set, T:X→X a bijection, and P its permutation matrix, with P[y,x]=1 when y=T(x). For 0<α<1, define Gα=(I−αP)^−1=Σj≥0 α^jP^j. If kx is the least positive integer such that T^kx(x)=x, then the exact diagonal return weight is gx=(Gα)[x,x]=(1−α^kx)^−1. Use gx normalized by its state average as a recurrent transport gain. The linear resolvent is well-defined because the spectral radius of αP equals α<1. For a fixed point and a period-k state, the gain gap is g1−gk=α(1−α^(k−1))/(1−α^k).
Implementation notes
1. Integration point: use a torus grid of N² tokens, a graph-neural network, or an SSM-like recurrent block. Choose an invertible integer matrix A modulo N, define T(x)=Ax modulo N, and transport features by gathering from T^−1(x). Compute all periods once with a visited-array cycle decomposition; for large spaces, estimate periods by bounded orbit simulation. Set α directly or parameterize it as sigmoid(a)(1−ε). 2. Pseudocode: periods = cycle_decompose(T); g = 1/(1-alpha**periods); g = g/mean(g); transport = gather(h, inverse_T); h = activation(W(h) + U(g[:,None]*transport) + c). Clip g when α is close to one. 3. Exact periods, gains, and the condition α<1 are computed from the paper; W, U, and c are learned. 4. First experiment: train this model and a matched message-passing baseline on modular-map next-state prediction and denoising for N=31 and N=64. Sweep α from 0.1 to 0.99 and measure gain by orbit class. The predicted signature is gx=1/(1−α^kx) within numerical tolerance, with the unique fixed point having maximal gain. The measured fixed-point-to-period-k gap should match the stated formula within 5 percent. Linear amplification should remain bounded for α<1 and become singular at α=1 before nonlinear effects are introduced.
Verification
This idea has not been verified yet.
Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.
Artifacts
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